Q14(b). Particle in a box with sides a, b, c: ψ, E and the Laplacian
Consider a particle moving in a three-dimensional box with sides a, b and c. Assuming n₁, n₂ and n₃ as quantum numbers for the motions along x, y and z directions, write down the wavefunction and energy for this system. State whether the pair of Laplacian operator and this wavefunction satisfy the eigenvalue equation. If yes, what is the eigenvalue?
Answer in brief
ψ = √(8/abc) sin(n₁πx/a) sin(n₂πy/b) sin(n₃πz/c), E = (h²/8m)(n₁²/a² + n₂²/b² + n₃²/c²); yes, ∇²ψ = λψ with λ = −8π²mE/h².
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Coversparticle in a three-dimensional boxLaplacian operatoreigenvalue equationseparation of variableswave function
All parts of Question 14
- 14(a) Degeneracy in a cubic box; states with E = 101h²/8mL²10 marks
- 14(b) Particle in a box with sides a, b, c: ψ, E and the Laplacian10 marks